Lemma 13.4.4.CA3A887474EE Let $Q\subset \mathbb C^ r$ and $D\subset \mathbb C^ m$ be open sets, $r,m\geq 1$, let $G\subset D$ be a Borel set, and for each $z\in G$ let $\mu _ z$ be a positive Radon measure on $Q$. Assume that $z\mapsto \int a\, \mathrm d\mu _ z$ is Borel on $G$ for every continuous function $a$ with compact support in $Q$. Then $z\mapsto \mu _ z(E_ z)$ is Borel on $G$ for every Borel set $E\subset Q\times D$, where $E_ z=\{ x:(x,z)\in E\} $. Moreover, for every nonnegative Borel function $f$ on $D$,
\[ \lambda (E)=\int _ G\mu _ z(E_ z)f(z)\, \mathrm dV(z) \]
is a measure on the Borel sets of $Q\times D$, and
\[ \int h\, \mathrm d\lambda =\int _ G\Bigl (\int _ Qh(x,z)\, \mathrm d\mu _ z(x)\Bigr )f(z)\, \mathrm dV(z) \]
for every nonnegative Borel function $h$ on $Q\times D$, the inner integral being a Borel function of $z\in G$.
Proof.
Fix open sets $Q''\Subset Q$ and $D''\Subset D$, so that $\mu _ z(Q'')<\infty $ for every $z$. If $A\subset Q''$ is open, the continuous functions $a_ l(x)=\min \{ 1,\max \{ l\operatorname {dist}(x,\mathbb C^ r\setminus A)-1,0\} \} $ have compact support in $A$ and increase to $\mathbf1_ A$, so $z\mapsto \mu _ z(A)$ is Borel by monotone convergence. Let $\mathcal D$ be the family of Borel sets $A\subset Q''$ for which $z\mapsto \mu _ z(A)$ is Borel on $G$. It contains the open subsets of $Q''$, which are closed under finite intersections and generate the Borel $\sigma $-algebra of $Q''$. It is closed under complements in $Q''$, because $\mu _ z(Q''\setminus A)=\mu _ z(Q'')-\mu _ z(A)$ with finite values, and under countable disjoint unions. By Dynkin's theorem (lemma 2.2.6), $\mathcal D$ contains every Borel subset of $Q''$.
Next, let $\mathcal D'$ be the family of Borel sets $E\subset Q''\times D''$ for which $z\mapsto \mu _ z(E_ z)$ is Borel on $G$. It contains the products $A\times A'$ of Borel sets $A\subset Q''$ and $A'\subset D''$, since $\mu _ z((A\times A')_ z)=\mu _ z(A)\mathbf1_{A'}(z)$; these products are closed under finite intersections and generate the Borel $\sigma $-algebra of $Q''\times D''$, because every open subset of $Q''\times D''$ is a countable union of products of open balls. As before, $\mathcal D'$ is closed under complements and countable disjoint unions, so it contains every Borel subset of $Q''\times D''$. Exhausting $Q$ and $D$ by increasing sequences of such sets $Q''$ and $D''$, and using monotone convergence, we obtain the first assertion.
By the first assertion, $\lambda (E)$ is defined, and $\lambda $ is countably additive by monotone convergence, applied first to the measures $\mu _ z$ and then to the integral over $G$. The formula for $\int h\, \mathrm d\lambda $ holds for indicator functions of Borel sets by definition, hence for nonnegative simple functions by linearity, and for nonnegative Borel functions by monotone convergence, applied in the same order; the inner integrals are Borel in $z$ as increasing limits of Borel functions.
$\square$
Pome
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